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A generalized Exner equation for sediment mass balance

A generalized Exner equation for sediment mass balance

A generalized Exner equation for sediment mass

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, F04014, doi:10.1029/2004JF000274, 2005 A generalized Exner equation for sediment mass balance C. Paola 1 and V. R. Voller 2 St. Anthony Falls Laboratory, Minneapolis, Minnesota, USA Received 13 December 2004; revised 7 June 2005; accepted 6 September 2005; published 30 November 2005. [1] The advance of morphodynamics research into new areas has led to a proliferation of forms of sediment mass balance equation. Without a general equation it is often difficult to know what these problem-specific versions of sediment mass balance leave out. To address this, we derive a general form of the standard Exner equation for sediment mass balance that includes effects of tectonic uplift and subsidence, soil formation and creep, compaction, and chemical precipitation and dissolution. The complete equation, (17), allows for independent evolution of two critical interfaces: that between bedrock and sediment or soil and that between sediment and flow. By eliminating terms from the general equation it is straightforward to derive mass balance equations applicable to a wide range of problems such as short-term bed evolution, basin evolution, bedrock uplift and soil formation, and carbonate precipitation and transport. Dropping terms makes explicit what is not being considered in a given problem and can be done by inspection or by a formal scaling analysis of the terms. Scaling analysis leads directly to dimensionless numbers that measure the relative importance of terms in the equation, for example, the relative influence of spatial versus temporal changes in sediment load on bed evolution. Combining scaling analysis with time averaging shows how the relative importance of terms in the equation can change with timescale; for example, the term representing bed evolution due to temporal change in sediment load tends to zero as timescale increases. Citation: Paola, C., and V. R. Voller (2005), A generalized Exner equation for sediment mass balance, J. Geophys. Res., 110, F04014, doi:10.1029/2004JF000274. 1. Introduction [2] An analysis of sediment mass balance is fundamental to solving a wide range of problems in morphodynamics. The relations in common use for expressing sediment mass balance take a variety of forms but are all descended from the equation initially presented by Exner and reproduced below. The additions and changes that have been made to Exner’s original equation have mostly been done piecemeal with the aim of adapting it for a particular problem. A summary of the Exner equation including forms appropriate for channelized systems and sediment mixtures is presented by Parker [2005], and detailed derivations of mass balance for soil-mantled hillslopes are presented by Anderson [2002] and Mudd and Furbish [2004]. Both of the latter works also explicitly include chemical effects. Nonetheless, each of these mass balance equations includes some terms and leaves out others, according to the problem at hand. Here we revisit the Exner equation with the aim of providing a complete general form. This generality allows the equation to be specialized for application to a wide range of morphodynamic problems by dropping or combining terms. 1 Also at Department of Geology and Geophysics, University of Minnesota Twin Cities, Minneapolis, Minnesota, USA. 2 Also at Department of Civil Engineering, University of Minnesota Twin Cities, Minneapolis, Minnesota, USA. Copyright 2005 by the American Geophysical Union. 0148-0227/05/2004JF000274$09.00 We hope that the proposed general form will be especially useful for geologic problems for which processes such as tectonic uplift and subsidence, soil formation and creep, and dissolution and precipitation become important. [3] It is easy to overlook the fact that a model is defined as much by what it leaves out as by what it includes. Using a general equation as a starting point makes this explicit: we develop specialized relations for specific problems by eliminating terms from the general equation. Elimination of terms can be done by inspection or by formal scaling analysis. We give examples of how to estimate the magnitudes of different terms below, after deriving the general mass balance equation. Because surface evolution occurs on timescales from seconds to millions of years, we then investigate the behavior of the general equation under time averaging. We conclude the paper by returning to one of the original motivations for it: the satisfaction of finding unity in apparently diverse phenomena. We hope that the examples that conclude the paper, showing how the general mass balance equation can be specialized to a variety of problems from classical morphodynamics to basin dynamics and carbonate precipitation, will illustrate how sediment mass balance can serve as one such unifying theme in surface dynamics. 2. Exner’s Equation [4] Felix Exner was a Viennese meteorologist who worked on a variety of topics in natural science. He F04014 1of8

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